Abstract: The inertia space of a Lie groupoid encodes interesting topological, geometric, and analytic information about the original Lie groupoid. It is the goal of the talk to explain this point of view using as example the cyclic homology theory of the convolution algebra of a proper Lie groupoid. To this end, the inertia groupoid associated to a proper Lie groupoid is first defined. We show that it is a differentiable stratified groupoid, and non-singular only in exceptional cases. The corresponding quotient space, the inertia space, possesses a Whitney stratification, and is triangulable. Finally, horizontal and basic forms over the inertia space are constructed, and a Hochschild-Kostant-Rosenberg type theorem for the convolution algebra of a proper Lie groupoid is indicated. The talk is based upon joint work, partially in progress, with H. Posthuma and X. Tang, as well as with C. Farsi and Ch. Seaton.